In [1]}, a geometric characterization of planar \((W^{1, 2}, W^{1, 2})\) -extension domain was given. By later results in [2]}, [3]}, [4]}, [5]}, we now have geometric characterizations of planar simply connected \((W^{1,p}, W^{1, p})\) -extension domains for all \(1\le p\le \infty \) . A geometric characterization is also known for planar simply connected \((L^{k, p}, L^{k, p})\) -extension domains with \(2<p\le \infty \) , see [6]}, [7]}, [8]}. Here \(L^{k, p}({\Omega })\)  denotes the homogeneous Sobolev space which contains locally integrable functions whose \(k\) -th order distributional derivative is \(L^p\) -integrable. Beyond the planar simply connected case, geometric characterizations of Sobolev extension domains are still missing. However, several necessary properties have been obtained for general Sobolev extension domains.
